Forward guidance and the term structure in New Keynesian models

Take a standard New Keynesian macroeconomic model, where the central bank sets the one-period interest rate. Now let's hit it with an unexpected shock. For simplicity and concreteness, let the shock be a reduction in government spending that will last for n periods, after which government spending will return to normal. So government spending was 100, will now be 90 for n periods, then will be 100 again in period n+1 and thereafter.

If n=1, the model says the central bank should cut the real interest rate for one period, then raise it again to its original level.

If n=2, the model says the central bank should leave the real interest rate unchanged for one period, then cut it in the second period, then raise it again to its original level in the third period.

If n=3, the model says the central bank should leave the real interest
rate unchanged for two periods, then cut it in the third period, then
raise it again to its original level in the fourth period.

Etc.

If n=infinity, the New Keynesian model says the central bank should leave the real interest rate unchanged.

My hunch is that most students of New Keynesian macroeconomics knew about the results for n=1 and n=infinity, but did not know about the results for n=2 and n=3 etc., and had to think about it. Am I right?

For the case where n=1, the New Keynesian model gives exactly the same policy advice as an Old Keynesian model. But for all the other cases it gives very different policy advice. That's because the Old Keynesian IS curve says that the level of private demand in period t is a negative function of the real interest rate in period t. But the New Keynesian IS curve says that the expected growth rate of private demand between periods t and period t+1 is a positive function of the real interest rate in period t.

How long is the period?

In the real world, central banks like the Bank of Canada set an overnight rate of interest, and they typically set it every six weeks. "Overnight" and "six weeks" are both very short periods. Most shocks last for longer than six weeks.

If the New Keynesian model is correct, it would be impossible for central banks to compensate for the effects of most shocks by doing what they do now. If a shock hit that was expected to last for 3 years, the central bank should use "forward guidance" and announce that it would lower the rate of interest temporarily 3 years from now. Or it could get to work on the term-structure of interest rates, by lowering the 3 year interest rate, while leaving all interest rates with less than 3 year terms unchanged.

I don't observe central banks actually doing anything like that.

The only sort of shocks where current policy would work would be shocks that have their maximal impact immediately, and then immediately start to decay at some geometric rate back to zero. In those special cases New Keynesian models would give the same advice, qualitatively, as Old Keynesian models.

If instead we had shocks that steadily grew over time, then started to decay back to zero, New Keynesian models would give the exact opposite advice to Old Keynesian models in the initial periods. For example, if the government announced that government spending would start to decline, stop declining after two years, then start to rise again, the New Keynesian model says the central bank should immediately raise real interest rates, slowly lower them for two years, then slowly start raising them again.

Do New Keynesians understand their own models? Do New Keynesians rig their models by only allowing shocks that make New Keynesian models behave like Old Keynesian models? Why aren't New Keynesian macroeconomists the ones arguing that central banks must operate on the whole term structure of interest rates and not just on the overnight rate of interest for the next 6 weeks?

Mathematic appendix: let C(t) be private demand, G(t) be government demand, and Y* be potential output, at time t. The Euler equation IS curve in New Keynesian models says that C(t)/Et[C(t+1)] = alpha(1+z)/(1+r(t)) where Et is expectations at time t, r(t) is the real interest rate, z is the rate of time preference proper, and alpha is a positive elasticity parameter. The central bank's job is to set r(t) such that C(t)+G(t)=Y*. Which means (Y*-G(t))/Et[Y*-G(t+1)] = alpha(1+z)/(1+r(t)). Assume C(n+1)+G(n+1)=Y* because that's what New Keynesians assume. Then solve backwards from there.

Peter N: there's a Calvo Phillips Curve that maps the output gap into inflation, and a definition of the nominal interest rate as real plus expected inflation, but you don't need them for this question.

In my opinion, it is definitely invalid to just assume the economy converges to potential output after the shock is past (and then solve backwards), as I've said in previous posts. But that's what they assume, so I'm just doing likewise. (I'm not sure if that's what you were asking; remember that C(t) can jump unexpectedly when new information arrives, so Et[C(t+1)] changes.)

Possibly confused here. Shouldn't the duration of a shock, once expected, be pulled forward into current demand (i.e. hurt current demand more than a shorter-duration expected shock), with the path of expected growth over the life of the shock then determined by the interest rate slash Euler equation and not the duration of the shock?

Peter N: if G is falling over time, then full employment requires C to be rising over time, which requires a high r in a New Keynesian model. Yes, it is counterintuitive if you come at it with Old Keynesian intuitions.

"Why aren't New Keynesian macroeconomists the ones arguing that central banks must operate on the whole term structure of interest rates and not just on the overnight rate of interest for the next 6 weeks?"

I guess you mean they should be saying things like this:

"Perhaps a complex offer by the central bank to buy and sell at stated prices gilt-edged bonds of all maturities, in place of the single bank rate for short-term bills, is the most important practical improvement which can be made in the technique of monetary management."

"The only sort of shocks where current policy would work would be shocks that have their maximal impact immediately, and then immediately start to decay at some geometric rate back to zero"

What's *a* shock? Let's say that individual shocks are of exactly this type and no other type. However, let's say there's a shock in every period, with *initial* amplitude +ve, -ve or zero. Let Si be the shock of period i, and Si(j) be the amplitude of that shock in period j, for all periods j>i.

So in this model, Si(i)>=Si(i+1)>=Si(i+2).....

However, S1(1), S2(2), S3(3) can be any amplitude, so that the total perturbation of period t = S1(t) + S2 (t) +.... + S3(t) can take any smooth continuous shape.

Now does the standard central banking recommendation of NK models work?

Or, if I have a model where the entire term structure moves with the short rate, does the NK model work?

Assume (just to make it easier for me) that all shocks are shocks to G.

A "shock at time t" means the central bank (and the population) gets *new information at time t* about G(t), and/or G(t+1) and/or G(t+2) etc.

Suppose for example that G(t) = some constant + S(t), where S(t) = B.S(t-1) + v(t) where 0 < B < 1 and v(t) is serially uncorrelated. Then I think setting the one period interest rate would work fine. The rest of the term structure would follow along, if markets knew the central bank was responding correctly and would continue to respond correctly in future.

It looks like NK models are sensitive to the central bank target function and there are a lot of similar NK models. To save us (me at least) a lot of aggravating math, what is your model's Euler equation for Y(t) and how does your objective function compare to

Peter N: In my example, I have assumed that the central bank learns about shocks the instant they happen, and can respond instantly too, and that there are no price shocks. So my central bank is able to minimise that same loss function you write down above, and minimise it at zero (inflation is always on target and Y is always at Y*).

The nominal interest rate rule can be found by solving (Y*-G(t))/Et[Y*-G(t+1)] = alpha(1+z)/(1+r(t)) for r(t), then adding target inflation.

Ralph: make it a temporary *increase* in government spending if you like, to cope with a sudden temporary emergency. Nothing changes, except the signs. Or make it a temporary drop on private demand if you like. Nothing changes, except it's harder for me to describe in words. Or it could be net exports, but then I have to talk about exchange rates and capital mobility, which makes it harder for me.

"and minimize [the loss function] at zero (inflation is always on target and Y is always at Y*)"

Thanks, this helps. As usual you've used interesting and radical simplification to highlight the problem. The question (for me, at least) is at what step in constructing the NK model was this behavior introduced.

The NK model makes any number of counter-factual assumptions (I counted over a dozen before I gave up), as all these models do, but many of them seem harmless. I thought it might be something with your target (since some choices of target produce multiple equilibria and other problems, you couldn't really solve backwards usefully), but now that you've clarified things for me, I think your simplified version is stable and the problem has to be earlier.

Naively I'd expect the growth rate of private demand between periods t and period t+1 to be a function of both the real interest rate in period t and an additional variable.

There's also the problem that all of our data comes from periods of central bank discretion.

BTW you can do Greek letters (under Windows) if you create a Greek keyboard mapping. Unfortunately proper subscripts and superscripts and an integral sign seem to be a bit harder.

It may matter that companies behave differently depending on the expected duration of a fall in demand. The longer the duration, the more sense it makes to retire older capacity without replacing it immediately or at all. This is going to have an effect on the output gap.

It's not clear to me how the NK model can deal with structural change based on expected demand. With such changes, doesn't backwards solution become a bit problematic?

Another shortcoming with the new keynesian model is that the central bank cant lower the real interest rate. It can lower the real interest rate on reserves but its not like the central bank directly lends reserves to the economy. For example if somehow the CB can increase inflations expectations by credibly promising to be irresponsible market interest rates will go up with the inflation. Market interest rates factor in inflation.

Nick, still probably confused, but why isn't it true that you are creating this period-constraint by back solving from an n+1 anchor period and thus fixing your expectations of the period-by-period C-at-a-given-real-rate, when nothing in the model requires this to be so? Irrespective of the expected duration of the actual shock (how long government spending is below normal or when the meteor hits), isn't the path of Y (or C) endogenous in a model that isn't back solving the path of r(t)? If a CB did decide to attempt to back solve to a given set of expectations, then it does seem like it could create a very counter-intuitive set of time varying real policy rates. But isn't there also an equilibrium solution across n periods where different r (different from the fixed back solving solution) at t1, t2, etc change the expectations about C in the nth period and thus changes the optimal nth period policy rate, but still creates a path to Y* at n+1? It seems like you are saying the NK model prevents the pulling forward of an expected shock in some sense when I wonder if you aren't just imposing this by back solving. But I'm wobbly here.

dlr: "Nick, still probably confused, but why isn't it true that you are creating this period-constraint by back solving from an n+1 anchor period and thus fixing your expectations of the period-by-period C-at-a-given-real-rate, when nothing in the model requires this to be so?"

You are *not* confused. It *is* true. And *nothing* in the model requires this to be so. It is *just* an assumption, and one that comes out of nowhere. And it sucks. But I made this assumption for *no* other reason than that NK models make this sucky assumption.

'Mike: if prices are sticky, and inflation is slow to adjust, real interest rates will change when monetary policy changes."

Real interest wont change much because an increase in inflation will also send market rates up. Lenders factor inflation into their lending. Market rates are relevant to the economy, not reserve rates.

Plus, please tell me: have you ever been taught New Keynesian macro models?

No. Whatever I think I get has come from doggy paddling through Interest and Prices a few times, reading slash trying to read lots of papers from Woodford, Gali, Gertler, Mankiw et al, a few conversations with some of current experts (could count as being taught) and then especially reading some of the criticisms from Buiter, Cochrane, McCallum, Colin Rogers. etc. And now you. I have read all three of the posts you linked and in general have found them incredibly eye opening, except I still don't trust my own eyelids on this stuff. There is so much out there on nominal indeterminacy in the canonical Woodford model but I don't see anyone talking about real indeterminacy the way you have, though maybe this is implicit in some of the multiple equilibria criticisms that just doesn't get laid out as cleanly as you have done it. I still don't *really* understand how these two potential indeterminacies relate. I would pay a lot of money to watch a panel with you, Cochrane and two serious NK trench diggers with good model-to-intuition translation skills go back and forth on these questions.

dlr: thanks for that response. You have done some serious reading in NK macro. The reason I asked is that, like you, I have been struggling over the years to really get my head around NK models, especially the Neo-Wicksellian (Woodfordian) variants where the central bank sets a nominal interest rate and we have an Euler equation IS curve. And I do not know how much other people know about what is really going on underneath the equations in NK models. For example, while Colin Rogers has been digging into the role of money as medium of account in NK models (that's my interpretation of Colin), I was previously digging into the role of money as medium of exchange in NK models. I think (I'm not sure) that some New Keynesians think that NK models work fine as models of barter economies. And I am sure they don't. They are models of monetary exchange economies without money!

I would really like someone like Simon Wren-Lewis to get involved in these discussions. I keep hoping he will rise to one of my baits, but he never has. Brad DeLong occasionally responds, and responds well, but Brad isn't really a New Keynesian (deep down I strongly suspect him of Monetarist sympathies!)

"Brad isn't really a New Keynesian (deep down I strongly suspect him of Monetarist sympathies!)"

Didn't he write a very good paper suggesting that those are one and the same thing, i.e. New Keynesianism could have justifiably been called New Monetarism?

If one were cynical, one would say that New Keynesianism is just a way that people who don't like Milton Friedman can accept the most of the monetarist position without using the nasty 'monetarist' label and instead using the name of cuddly Keynes?

Here's the paper. I haven't read it in a while, but I remember thinking that it was really good-

"The story of 20th century macroeconomics begins with Irving Fisher. In his books Appreciation and Interest (1896), The Rate of Interest (1907), and The Purchasing Power of Money (1911), Fisher fueled the intellectual fire that became known as monetarism. But what has happened to monetarism at the end of the 20th century? The short answer is that much of this current of thought is still there, but its insights pass under another name. We may not all be Keynesians now, but the influence of monetarism on how we all think about macroeconomics today has been deep, pervasive, and subtle. Why then do we today talk much more about the "New Keynesian" economists than about the "New Monetarist" economists? I believe that to answer this question we need to look at the history of monetarism over the 20th century. One fruitful way to look at the history of monetarism is to distinguish between four different variants or subspecies of monetarism that emerged and for a time flourished in the century just past: First Monetarism, Old Chicago Monetarism, High Monetarism, and Political Monetarism."

W Peden: That paper was published early 2000, and probably written 1998 or 1999. It's a paper about the 20th century, right from the very first line. Notice how Woodford's name is missing from the references. Because it was written just a little too early to catch the latest Neo-Wicksellian twist on New Keynesian, where money was dropped from the model. So I think it was right (and a very good paper) when written, but a revised paper written now would be different. Put it this way: I think that Brad still has money very much in his mental model, but New Keynesians don't any more. That makes him a monetarist (sort of), as opposed to a 21st century New Keynesian.